Equation of Cardioid

Theorem

Polar Equation

Let $C$ be a cardioid embedded in a polar coordinate plane such that:

its deferent of radius $a$ is positioned with its center at $\polar {a, 0}$

there is a cusp at the origin.

The polar equation of $C$ is:

$r = 2 a \paren {1 + \cos \theta}$

Parametric Equation

Let $C$ be a cardioid embedded in a Cartesian coordinate plane such that:

its deferent of radius $a$ is positioned with its center at $\tuple {a, 0}$

there is a cusp at the origin.

Then $C$ can be expressed by the parametric equation:

$\begin {cases} x = 2 a \cos t \paren {1 + \cos t} \\ y = 2 a \sin t \paren {1 + \cos t} \end {cases}$